Identification of Broadband Source-Array Responses from Sensor Second Order Statistics

Abstract

Abstract-This paper addresses the identification of sourcesensor impulse responses from the measured space-time covariance matrix in the absence of any further side information about the source or the propagation environment. Using polynomial matrix decomposition techniques, the responses can be narrowed down to an indeterminacy of a common polynomial factor. If at least two different measurements for a source with constant power spectral density are available, this indeterminacy can be reduced to an ambiguity in the phase response of the sourcesensor paths.

Mathematical Tools for Processing Broadband Multi-Sensor Signals

“…For these, algorithmic efforts have been reviewed, with the main focus currently directed to extracting analytic eigenvalues and -vectors that afford lower-order polynomial approximations than the state-of-the-art in PEVD approaches with proven convergence. Important future developments also target the estimation of space-time covariance [63,64,65], since the statistics typically have to be estimated from finite data sets, and interesting new applications where the polynomial approach permits solutions that were previously unobtainable, such as in the area of impulse response modelling [66] or speech enhancement [67].…”

Sample Space-time Covariance Matrix Estimation

“…A spectrally majorised, not necessarily analytic version of this factorisation is the McWhirter decomposition [3], which approximates the factorisation by polynomial paraunitary and diagonal parahermitian matrices. A number of algorithms for the latter have emerged [3][4][5][6][7][8][9][10] and in turn triggered various applications ranging from broadband multiple-input and multipleoutput (MIMO) systems [11,12], to coding [13], beamforming [14,15], source separation [16] and angle of arrival estimation [17,18], to name but a few.…”

Support Estimation of a Sample Space-Time Covariance Matrix

“…where E{•} is the expectation operator, represents the data's second order statistics, and is therefore central in the formulation of many broadband array processing problems. This includes for example broadband MIMO systems [1], coding [2], beamforming [3], [4], source separation [5], angle of arrival estimation [6], scene discovery [7], and many others applications. Based on factorisations of its z-transform R(z) = τ R[τ ]z −τ such as the polynomial eigenvalue (EVD) [8]- [10], or singular value decompositions [8], [11], well-known narrowband optimal solutions [12], [13] can be directly extended to the broadband case.…”